DOCSLIB.ORG

Working Paper

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Working Paper WORKING PAPER GRAPHICAL CONVERGENCE OF SET-VALUED MAPS Jean-Pierre Aubin September 1987 WP-87-083 International Institute for Applied Systems Analysis NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR GRAPHICAL CONVERGENCE OF SET-VALUED MAPS Jean-Pierre Aubin* September 1987 WP-87-083 * CEREYADE, Universitg de Paris-Dauphine, Paris, France Working papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations. INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A- 236 1 Laxenburg, Austria FOREWORD This is an introduction to graphical convergence of set,-valued maps and the epigraphical convergence of extended real-valued functions1. It is now well established that maps, and more generally, set-valued maps, should be regarded not only as maps from one space to another, but should be characterized in an intrinsic and symmetric way by their graphs. When dealing with limits of maps, either single-dued or set-valued, it is quite advantageous to wercome the natural reluctance to handle conver- gence of subsets and to replace pointwise convergence by "graphical con- vergence": Instead of studying (more or less uniform) limits of the images, one consider the limits of their graphs. One of the main reasons is that doing so is that a map and its inverse are treated on the same footing . This is quite important in approximation theory and numerical analysis. The concepts of graphical convergence of set-valued maps are related to the concepts of epigraphical limits of functions, which had recently met an important success to wercome the failure of pointwise convergence in many problems of calculus of variations, optimization, stochastic programming, etc. Finally, this report provides a first study of the Kuratowslii upper and lower limits of tangent cones, which is needed to compute general- ized derivatives and epi-derivatives of graphical and epigraphical limits of maps and functions. Alexander B. Kurzhanslri Chairman S?rstem and Decision Sciences Program - 'in the development of which IIASA played an important role. ... lll Introduction Here are some basic notes on graphical limits of single-tdued and/or set.-valued maps. To deal with limits is the basis of analysis, or approximation theory, where we have to approximate objects by richer and more familiar ones. And numerical analysis is just taking up this issue for very practical pur- poses. In particular, solving equations and approximating their solutions is the ultimate task of many mathematicians. It can be regarded in this way: We want to approximate a solution xo to an equation f (xo)= yo. by approximating both the data yo by a sequence of approximatte data ynas, and the map f by a sequence of maps fn. Knowing how to find solutions to the approximate problems fn(x4)= y,, the problem is how to derive the convergence of the approximate solu- tions from the convergence of the data y, and fn and thus, to pay some attention to the limits of maps. It is nowT well established that maps, and more generally, set-valued maps, should be regarded not only as maps from one space to another, but should characterized in an intrinsic and symmetric way by their graphs. This point of view, which goes back to the protohiston of analysis with Fermat and Descartes dealing with curves rather than functions, has been let aside since for a long time. In particular. as far as limits of functions and maps go, generations of mathematicians have be accustomed to deal with many concepts of convergence of functions, from pointwise to uniform, but all based on the fact that a map is a map, and not a graph. When dealing nit11 limits of maps, either single-valued or set-valued. it is quit,e advantageous to overcome the natural reluctance to handle conver- gence of subsets and to replace pointwise convergence by "graphical con- vergence': Instead of studying (more or less uniform) limits of the images, we consider the limits of their graphs2. One of the main reasons is that doing so is that we treat on the same footing a map and its inverse. This is quite important in approximation h his point of view of regarding maps as graphs has already allowed tn build a successful differential calculus of set-valued maps based on 'graphical derivatives'rather than set- valued limits of differential quotients. theorya, where the problem is to derive poinimise convergence of the in- verses from the point,mise convergence of the maps. Hence. a first problem is t,o study what are the relations between graph- ical and pointwise convergence. The main tool for that is the Stabiliq- Theorem4. an outgrowth of the Inverse Function Theorem for set-dued maps6. Since graphical limits are limit of graphs, which are subsets. we have to rely on the now classical concepts of Kuratowski upper and lower limits. Hence we begin by a short exposition of these concepts. With the Stability Theorem on one hand. and the concept of proper maps on the other, we are also able to complement the calculus of Kuratowski upper and lower limits of sets. and in particular. to protide crit,eria for having the natural formulas for direct and inverse images of Kuratowski limits. After defining and providing the basic properties of graphical conver- gence, we expose its applications to viability theory, where we show, for instance that the Kuratowski upper limit of viability domains is a viability domain of graphical upper limits. Finally, we relate the concepts of graphical convergence of set-valued maps to the concepts of epigraphical limits of functions, which has re- cently met an important success to overcome the failure of pointwise conver- gence in many problems of calculus of variations, optimization, sochastic programming6, etc. The use of this concept is mandatory whenever the order relation of the real line comes into play, as in optimization or Lyapunm- stability for instance. In such cases, a real-valued function is replaced by the set-valued map obtained by adding to it the positive cone (for minimization). whose graph is thus the epigraph of the function. Therefore. the convergence of the graphs of such set-valued maps is the convergence of the epigraphs of the associated functions. We present just a selection of issues dealing with epi-convergence, among 3see 112, Stability Theorem 1.11, which adapt to the general case of solving inclusions the principle stating that stability, convergence of the data and "consistency' imply the rnn- vergence of the solutions. Consistency is nothing other than graphical lower convergence. 4see [12, Proposition 1.11 &see[lo, Chapter 71 and jll] 6see for instance the book [S] and the bibliography of this took. which some formulas dealing with the epi-limits of sum and products of functions. Finally, we relate these concepts of Kuratowski limits with the ones of tangent conesi. which lay the foundations of the differential calculus of set- valued maps, and which play such an important role in optimization and viability theory. The problem we begin to consider is to stady the Iiuratowski upper and lower limits of the tangent cones to subsets K, in terms of Kuratowski upper and lower limits of sets of the form (K, - x,)/hn when h;, 3 x, - r and h, - O+. which we call respectively asymptotic paratingent and circatangent cone. One would like to relate them to the tangent cones t,o the Kuratowski limits of a sequence of subsets fin, but this seems rather difficult outside the convex realm. Let us just point out that Clarke tangent cones and Bouligand's paratingent cones to a subset K can be regarded as asymptotic circatangent and paratingent cones for constant sequences. 'B~the way. the various definitions of tangent cones are Kuratowski upper and lower limits of the sets (K - x)ih when 11 + o+ Contents 1 Kuratowski Upper and Lower limits 5 2 Kuratowski Limits in Lebesgue Spaces 10 3 Stability Theorem 11 4 Kuratowski Limits of Inverse Images 15 5 Kuratowski Limits of Direct Images 19 6 Graphical Limits 20 7 Stability of Viability Domains and Solution Maps 24 8 Epigraphical Limits 27 9 Epigraphical limits of Sums of Functions 36 10 At t ouch's Theorem 39 11 Asymptotic Paratingent and Circat angent Cones 42 12 Asymptotic Paratingent and Circat angent Epiderivat ives 48 1 Kuratowski Upper and Lower limits We can also characterize closed set?\dued maps and lower semicontinuous set-valued maps through adequate concepts of limits. For that purpose. we introduce the following notations: we associate with a set-valued map F : S .u E' and x E ,Y the subsets I i) lims~p,~,F(f) := {y E Y I liminf$,, d(y, F(xl)) = 0} ii ) lim inf,,, F(xl) 1 := {y E E' ( lim$,, d(y, F(xl))= 0} They are otniousl?; closed. We also see at once that (2) lim inf F (z') c F(z) c lim sup F (XI) .f +.T 9-3 The other advantage of introducing these notions is that we can define kinds of semicontinuity for "discrete" set-valued maps, i.e., "semin limits of sequences of subsets h;, of a metric space X: we set When K, is a sequence of subsets of a metric space S,we shall also use the following notations: \ i) Kb= lim sup,,, K, 1 ii) K\= liminf,,,, K,, Definition 1.1 When F : .X - Er ia a aet-valued map, we cay/ that lirn sup F(xl) 3'-r ia the Kuratowski (or Kuratowski-Painlevh ) upper limit of F(xl) when x1 -- x and that lim inf F (XI) 3'-3 is the Kuratonrslii for Kuratonrslii-Painle~ri) lower limit of F(xl) when x1 - x.
Load more

Recommended publications
  • 1 Introduction
    DC APPROACH TO REGULARITY OF CONVEX MULTIFUNCTIONS WITH APPLICATIONS TO INFINITE SYSTEMS1 B. S. MORDUKHOVICH 2 and T. T. A. NGHIA3 Abstract. The paper develops a new approach to the study of metric regularity and related well-posedness properties of convex set-valued mappings between general Banach spaces by reducing them to unconstrained minimization problems with objectives given as the difference of convex (DC) functions. In this way we establish new formulas for calculating the exact regularity bound of closed and convex multifunctions and apply them to deriving explicit conditions ensuring well-posedness of infinite convex systems described by inequality and equality constraints. Key Words: variational analysis, optimization, convex multifunctions, metric regularity AMS Classification: 49J53, 49J52, 90C25 1 Introduction Let X and Y be Banach spaces with norms generically denoted by k · k. For any x 2 X and r > 0 the symbol IBr(x) stands for the closed ball centered at x with radius r, while the unit closed ball and the unit sphere in X are denoted by IBX and SX , respectively. Recall that a set-valued mapping/multifunction F : X ! Y is metrically regular around (¯x; y¯) 2 gph F with modulus K > 0 if there exist neighborhoods U ofx ¯ and V ofy ¯ such that dx; F −1(y) ≤ Kdy; F (x) for any x 2 U; y 2 V; (1.1) where d(x; Ω) stands for the distance from x to Ω in X. The infimum of all the moduli K over (K; U; V ) in (1.1), denoted by reg F (¯x; y¯), is called the exact regularity bound.
    [Show full text]
  • An Inverse Mapping Theorem for Set-Valued Maps F Acting from a Complete Metric Space I Toa Linear Space Y with a (Translation) Invariant Metric
    proceedings of the american mathematical society Volume 121, Number 2, June 1994 AN INVERSE MAPPING THEOREM FOR SET-VALUEDMAPS A. L. DONTCHEVAND W. W. HAGER (Communicated by James West) Abstract. We prove that certain Lipschitz properties of the inverse F-1 of a set-valued map F are inherited by the map (f+F)~x when / has vanishing strict derivative. In this paper, we present an inverse mapping theorem for set-valued maps F acting from a complete metric space I toa linear space Y with a (translation) invariant metric. We prove that, for any function f: X -> Y with "vanishing strict derivative", the following properties of the inverse map F~x are inherited by (f + F)~x : F~x has a closed-valued pseudo-Lipschitz selection, F~~x is locally closed-valued and pseudo-Lipschitz, F~x has a Lipschitz selection, and F~x is locally single-valued and Lipschitz. Let us recall that the distance from a point x to a set A in a metric space (X, p) is defined by dist(x, A) = inf{p(x, y) : y e A} and the excess e from the set A to the set B (also called the Hausdorff semidis- tance from B to A) is given by e(B, A) = sup{dist(x, A): x e B}. We denote by Ba(x) the closed ball centered at x with radius a. For notational convenience, t500(x) denotes X. Also recall that in a linear metric space (Y, d) the metric d is invariant if it satisfies d(x + z, y + z) = d(x, y) for every x,y, ze Y.
    [Show full text]
  • Quantitative Stability Analysis of Stochastic Quasi-Variational Inequality Problems and Applications
    Quantitative Stability Analysis of Stochastic Quasi-Variational Inequality Problems and Applications Jie Zhang1 Huifu Xu2 and Li-wei Zhang3 December 30, 2014 Abstract. We consider a parametric stochastic quasi-variational inequality problem (SQVIP for short) where the underlying normal cone is defined over the solution set of a parametric stochastic cone system. We investigate the impact of variation of the probability measure and the parameter on the solution of the SQVIP. By reformulating the SQVIP as a natural equation and treating the orthogonal projection over the solution set of the parametric stochastic cone system as an optimization problem, we effectively convert stability of the SQVIP into that of a one stage stochastic program with stochastic cone constraints. Under some moderate conditions, we derive H¨olderouter semicontinuity and continuity of the solution set against the variation of the probability measure and the parameter. The stability results are applied to a mathematical program with stochastic semidefinite constraints and a mathematical program with SQVIP constraints. Key words. Stochastic quasi-variational inequality, quantitative stability analysis, mathematical program with stochastic semidefinite constraints, mathematical program with SQVIP constraints AMS Subject Classifications. 90C15, 90C30, 90C33. 1 Introduction Let X, Y and Z be Banach spaces equipped with norm k·kX ; k·kY and k·kZ respectively. For x0 2 X and a closed convex set Y ⊂ Y , we consider the following stochastic quasi-variational inequality problem
    [Show full text]
  • On Metric Subregularity for Convex Constraint Systems by Primal
    Optim Letter manuscript No. (will be inserted by the editor) On Metric Subregularity for Convex Constraint Systems by Primal Equivalent Conditions Liyun Huang · Zhou Wei Received: date / Accepted: date Abstract In this paper, we mainly study metric subregularity for a convex con- straint system defined by a convex set-valued mapping and a convex constraint subset. The main work is to provide several primal equivalent conditions for metric subregularity by contingent cone and graphical derivative. Further it is proved that these primal equivalent conditions can characterize strong basic constraint qual- ification of convex constraint system given by Zheng and Ng [SIAM J. Optim., 18(2007), pp. 437-460]. Keywords primal condition · metric subregularity · contingent cone · graphical derivative · convex constraint system Mathematics Subject Classification (2010) 90C31 · 90C25 · 49J52 · 46B20 1. Introduction Many optimization problems appearing in variational analysis and mathematical programming can be modelled as finding a solution to a generalized equation. This generalized equation mathematically is defined as follows y¯ ∈ F (x) (1.1) where F : X ⇒ Y is a set-valued mapping between Banach spaces X and Y andy ¯ is a given point in Y . This research was supported by the National Natural Science Foundation of P. R. China (grant 11401518), the Fok Ying-Tung Education Foundation (grant 151101) and the Scientific Research Foundation from Education Department of Yunnan Province under grant 2015Z009. Liyun Huang School of Mathematics and Information Science, Qujing Normal University, arXiv:1610.05671v1 [math.OC] 18 Oct 2016 Qujing 655011,Yunnan Province, People’s Republic of China E-mail: [email protected] Zhou Wei( ) Department of Mathematics, Yunnan University, Kunming 650091, People’s Republic of China E-mail: [email protected] 2 Liyun Huang, Zhou Wei It is well known that a key concept, when studying the behavior of the solution set to generalized equation (1.1), is metric regularity.
    [Show full text]
  • Metric Regularity and Systems of Generalized Equations
    Metric regularity and systems of generalized equations Andrei V. Dmitruk a, Alexander Y. Kruger b,∗ aCentral Economics & Mathematics Institute, RAS, Nakhimovskii prospekt 47, Moscow 117418, Russia bSchool of Information Technology and Mathematical Sciences, University of Ballarat, POB 663, Ballarat, Vic 3350, Australia Abstract The paper is devoted to a revision of the metric regularity property for mappings between metric or Banach spaces. Some new concepts are introduced: uniform metric regularity and metric multi-regularity for mappings into product spaces, when each component is perturbed independently. Regularity criteria are established based on a nonlocal version of Lyusternik-Graves theorem due to Milyutin. The criteria are applied to systems of generalized equations producing some “error bound” type estimates. Key words: Variational analysis, Lyusternik-Graves theorem, regularity, set-valued mapping, constraint qualification 1 Introduction The property of metric regularity has proved to be one of the central concepts of the contemporary variational analysis, playing an extremely important role both in theory and its numerous applications to generalized equations, variational in- equalities, optimization, etc. Being well-established and recognized, this concept still continues its expansion into new areas of mathematical analysis (see the recent monographs [19,22] and survey papers [2,13]). ∗ Corresponding author. Email addresses: [email protected] (Andrei V. Dmitruk ), [email protected] (Alexander Y. Kruger). URL: http://uob-community.ballarat.edu.au/~akruger (Alexander Y. Kruger). Preprint submitted to Journal of Mathematical Analysis and Applications 7 February 2008 A set-valued mapping F : X ⇒ Y between metric spaces is said to be metrically regular near (x◦, y◦) ∈ gph F if there exist κ ≥ 0 and δ > 0 such that −1 ◦ ◦ d(x, F (y)) ≤ κd(y, F (x)), ∀x ∈ Bδ(x ), ∀y ∈ Bδ(y ).
    [Show full text]
  • Implicit Multifunction Theorems
    Implicit Multifunction Theorems Yuri S. Ledyaev1 and Qiji J. Zhu2 Department of Mathematics and Statistics Western Michigan University Kalamazoo, MI 49008 Abstract. We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characteriza- tion of its coderivative. Traditional open covering theorems, stability re- sults, and su±cient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson-Ursescu open mapping theorem. As a tool for this al- ternative proof of the Robinson-Ursescu theorem we also establish a re¯ned version of the multidirectional mean value inequality which is of independent interest. Key Words. Nonsmooth analysis, subdi®erentials, coderivatives, implicit function theorem, solvability, stability, open mapping theorem, metric reg- ularity, multidirectional mean value inequality. AMS (1991) subject classi¯cation: 26B05. Note. This is a update on July 18, 2006. We corrected an inaccuracy in the proof of Theorem 2.7 existed in the version published in Set Valued Analysis, 7: 209-238, 1999. We thank Professor W. Schirotzek for brought our attention to this inaccuracy. 1Research partially supported by the Russian Fund for Fundamental Research Grant 96-01-00219, on leave from Steklov Institute of Mathematics. 2Research partially supported by the National Science Foundation under grant DMS- 9704203 and by the Faculty Research and Creative Activities Support Fund at Western Michigan University. 1 Introduction In this paper we prove a general implicit multifunction theorem for the inclusion 0 2 F (x; p) (1) where F : X £ P ! 2Y is a multifunction, X and Y are Banach spaces and P is a metric space.
    [Show full text]
  • Implicit Multifunction Theorems in Complete Metric Spaces
    Implicit Multifunction Theorems in complete metric spaces Huynh Van Ngai ∗ Nguyen Huu Trony and Michel Th´era z Abstract In this paper, we establish some new characterizations of the metric regularity of implicit mul- tifunctions in complete metric spaces by using the lower semicontinuous envelopes of the distance functions for set-valued mappings. Through these new characterizations it is possible to investigate implicit multifunction theorems based on coderivatives and on contingent derivatives as well as the perturbation stability of implicit multifunctions. Mathematics Subject Classification: 49J52, 49J53, 90C30. Key words: Error bound, Perturbation stability, Metric regularity, Implicit multifunction, Gen- eralized equations. 1 Introduction Let X and Y be metric spaces endowed with metrics both denoted by d(·; ·): The open ball with center x and radius r > 0 is denoted by B(x; r): We recall that a set-valued (multivalued) mappping F : X ⇒ Y is a mapping which assigns to every x 2 X a subset (possibly empty) F (x) of Y . As usual, we use the notation gph F := f(x; y) 2 X × Y : y 2 F (x)g for the graph of F , Dom F := fx 2 X : F (x) 6= ;g −1 for the domain of F and F : Y ⇒ X for the inverse of F . This inverse (which always exists) is defined by F −1y := fx 2 X : y 2 F (x)g; y 2 Y and satisfies (x; y) 2 gph F () (y; x) 2 gph F −1: It is well known that a large amount of problems, such that for instance, inequalities and equalities systems, variational inequalities or systems of optimality conditions are covered by the solvability of a generalized equation (in the terminology of Robinson): For a given y 2 Y , determine x 2 X such that y 2 F (x).
    [Show full text]
  • Quantitative Stability Analysis of Stochastic Quasi-Variational Inequality Problems and Applications
    Quantitative Stability Analysis of Stochastic Quasi-Variational Inequality Problems and Applications Jie Zhang1 Huifu Xu2 and Li-wei Zhang3 May 30, 2015 Abstract. We consider a parametric stochastic quasi-variational inequality problem (SQVIP for short) where the underlying normal cone is defined over the solution set of a parametric stochastic cone system. We investigate the impact of variation of the probability measure and the parameter on the solution of the SQVIP. By reformulating the SQVIP as a natural equation and treating the orthogonal projection over the solution set of the parametric stochastic cone system as an optimization problem, we effectively convert stability of the SQVIP into that of a one stage stochastic program with stochastic cone constraints. Under some moderate conditions, we derive H¨olderouter semicontinuity and continuity of the solution set against the variation of the probability measure and the parameter. The stability results are applied to a mathematical program with stochastic semidefinite constraints and a mathematical program with SQVIP constraints. Key words. Stochastic quasi-variational inequality, quantitative stability analysis, mathematical program with stochastic semidefinite constraints, mathematical program with SQVIP constraints AMS Subject Classifications. 90C15, 90C30, 90C33. 1 Introduction Let X, Y and Z be Banach spaces equipped with norm k·kX ; k·kY and k·kZ respectively. For x0 2 X and a closed convex set Y ⊂ Y , we consider the following stochastic quasi-variational inequality problem (SQVIP):
    [Show full text]
  • A Unified Theory for Metric Regularity of Multifunctions
    Journal of Convex Analysis Volume 13 (2006), No. 2, 225–252 A Unified Theory for Metric Regularity of Multifunctions D. Az´e UMR CNRS MIP, Universit´ePaul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex, France Dedicated to Jean-Pierre Aubin on the occasion of his 65th birthday. Received: March 30, 2005 Revised manuscript received: June 16, 2005 We survey a large bunch of results on metric regularity of multifunctions that appeared during the last 25 years. The tools used for this survey rely on a new variational method introduced in [9]1 and further developped in [10] and independently in [35, 36] which provides a characterization of metric regularity. It allows us to give a unified point of view both for primal results (based on some notion of tangent cones) and dual ones (based on some notion of normal cones). For most known results on metric regularity, a simple proof is given along with some slight improvements for some of them. 1. Introduction The theory of metric regularity, whose origin dates back to [44] and [28], for mappings and further for multimappings is one of the cornerstones of nonsmooth analysis. A de- cisive step has been done in this theory by the systematic use of Ekeland's variational principle. However, there are some cases such as the Ursescu-Robinson Theorem and the finite dimensional case in which one can give metric regularity results without using the variational principle. We refer to [36] for its accurate bibliographical and historical comments. The first time in which these kinds of techniques has been used for metric regularity of mappings and multimappings seems to be [31] and [1].
    [Show full text]
  • A. L. Dontchev, W. W. Hager, and V. M. Veliov
    Set-Valued Analysis 8: 31–50, 2000. 31 © 2000 Kluwer Academic Publishers. Printed in the Netherlands. On Quantitative Stability in Optimization and Optimal Control ? A. L. DONTCHEV1,W.W.HAGER2, K. MALANOWSKI3 andV.M.VELIOV4 1 Mathematical Reviews, Ann Arbor, MI 48107, U.S.A. e-mail: [email protected]. 2 Department of Mathematics, University of Florida, Gainesville, FL 32611, U.S.A. e-mail: [email protected]fl.edu, http://www.math.ufl.edu/Qhager. 3 Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warsaw, Poland. e-mail: [email protected]. 4 Institute of Mathematics and Informatics, Bulgarian Acad. of Sc., 1113 Sofia, Bulgaria and Vienna University of Technology, Wiedner Hauptstr. 8-10/115, A-1040 Vienna, Austria. e-mail: [email protected]. (Received: July 1999) Abstract. We study two continuity concepts for set-valued maps that play central roles in quan- titative stability analysis of optimization problems: Aubin continuity and Lipschitzian localization. We show that various inverse function theorems involving these concepts can be deduced from a single general result on existence of solutions to an inclusion in metric spaces. As applications, we analyze the stability with respect to canonical perturbations of a mathematical program in a Hilbert space and an optimal control problem with inequality control constraints. For stationary points of these problems, Aubin continuity and Lipschitzian localization coincide; moreover, both properties are equivalent to surjectivity of the map of the gradients of the active constraints combined with a strong second-order sufficient optimality condition. Mathematics Subject Classifications (2000): 47H04, 90C31, 49K40.
    [Show full text]
  • International Journal of Pure and Applied Mathematics ————————————————————————– Volume 47 No
    International Journal of Pure and Applied Mathematics ————————————————————————– Volume 47 No. 3 2008, 325-333 METRIC REGULARITY AND SUBDIFFERENTIAL OF A FUNCTION AT A POINT FOR A CLASS OF FUNCTIONS Alexandru V. Blaga Department of Mathematics National College “Mihai Eminescu” 5 Mihai Eminescu Street, Satu Mare, 440014, ROMANIA e-mail: [email protected] Abstract: The property of metric regularity of a multifunction can be empha- tized in several ways. The Frechet differential at a point, the strict differential at a point or the strict slope at a point of a function can be directly linked to metric regularity. The condition which involve the metric regularity are men- tioned in [9], [5], [8] and [2]. This article emphasizes the connection between the subdifferential of a function at a point and the metric regularity for new multifunctions. AMS Subject Classification: 49J52, 58C20, 47D15, 49J50 Key Words: Banach spaces, Frechet derivative, metric regularity, Lyusternik- Graves Theorem, Robinson-Ursescu Theorem, coderivatives, slope of a function, subdifferentials, subgradient 1. Preliminaries There are several equivalent definitions of metric regularity on Banach spaces. The concept has been presented in many basic papers and articles. The the- ory of metric regularity is an extension of two classical results: the Lyusternik Theorem and the Graves surjection functions Theorem. Developments in non- smooth analysis in the 1980s and 1990s paved the way for a number of far- reaching extensions of these results. The well-known Lyusternik-Graves Theo- Received: March 13, 2008 c 2008, Academic Publications Ltd. 326 A.V. Blaga rem assures the property of metric regularity for differentiable functions at each point from the graph, proving through the equality reg F (¯x|y¯) = reg DF (¯x) ex- actly the modulus of metric regularity [2].
    [Show full text]
  • Samir Adly , Florent Nacry and Lionel Thibault
    ESAIM: Control, Optimisation and Calculus of Variations URL: http://www.emath.fr/cocv/ PROX-REGULARITY APPROACH TO GENERALIZED EQUATIONS AND IMAGE PROJECTION. Samir Adly1, Florent Nacry1 and Lionel Thibault2; 3 Abstract. In this paper, we first investigate the prox-regularity behaviour of solution mappings to generalized equations. This study is realized through a nonconvex uniform Robinson-Ursescu type theorem. Then, we derive new significant results for the preservation of prox-regularity under various and usual set operations. The role and applications of prox-regularity of solution sets of generalized equations are illustrated with dynamical systems with constraints. AMS Subject Classification. 49J52, 49J53. The dates will be set by the publisher. Contents 1. Introduction 2 1.1. Context of the paper. 2 1.2. Contributions of the paper. 3 1.3. Organization of the paper. 3 2. Application to nonsmooth dynamics 3 3. Notation and preliminaries 6 4. Prox-regularity and !(·)-normal regularity concepts 8 5. Metric regularity via prox-regularity/normal !(·)-regularity 11 6. Prox-regularity of the solution mappings to generalized equations 18 7. Preservation of the prox-regularity under various set operations 23 8. Concluding remarks 32 References 32 Keywords and phrases: Variational analysis; prox-regular set; metric regularity; generalized equation; Robinson-Ursescu Theorem; variational inclusion; nonsmooth dynamics. 1 Laboratoire XLIM, Universit´ede Limoges, 123, Avenue Albert Thomas, 87060 Limoges, CEDEX, France; e-mail: [email protected] & [email protected] 2 D´epartement de Math´ematiques,Universit´eMontpellier, 34095 Montpellier CEDEX 5, France; e-mail: [email protected] 3 Centro de Modelamiento Matematico, Universidad de Chile c EDP Sciences, SMAI 1999 2 PROX-REGULARITY APPROACH TO GENERALIZED EQUATIONS 1.
    [Show full text]